The two smallest commas I have found so far

So, get ready, this seems to be "something".

#1:

[-48 0 11 8]

Being tempered out in mohajira, this obviously suggests a "threeless" 2D system reaching 5s using 8 generators and 7s using -11 generators. Splitting the quarter-comma meantone fifth in two steps of 5^(1/8) is therefore an excellent idea and 7/4 is mapped literally to a "semidiminished 7th". This would be the best way of defending Vicentino's semichromatic steps that I have ever thought of. It even makes me speculate whether Vicentino was doing it really just for the sake of splitting the chroma and whether he hadn't actually considered involving 7/4s (without having told anyone).

#2:

[-1 -11 -1 0 6]

This suggests a "sevenless" 3D temperament of unmatched accuracy and rarely low complexity in the realm of microtemperaments. I'm only not sure about the generators right now.

Petr

2011/7/11 Petr Pařízek <petrparizek2000@...>
>
> #2:
>
> [-1 -11 -1 0 6]
>
> This suggests a "sevenless" 3D temperament of unmatched accuracy and rarely
> low complexity in the realm of microtemperaments. I'm only not sure about
> the generators right now.

In the 5-limit, this is interesting; it continues the pattern magic,
wurrschmidt, parizek#2. Magic combines the chain of thirds with 3/1,
Wurrschmidt does it with 6/1, this seems to do it with 12/1.

-Mike

Mike wrote:

> In the 5-limit, this is interesting; it continues the pattern magic,
> wurrschmidt, parizek#2. Magic combines the chain of thirds with 3/1,
> Wurrschmidt does it with 6/1, this seems to do it with 12/1.

The 11th root of 12/1, if that's what you're talking about, is remarcably wider than 5/4 and therefore I don't see much of a point in such a temperament, concerning its complexity. For that reason, I'm not surprised I haven't seen this temperament mentioned anywhere.

How have you found those patterns?

BTW: By "parizek#2", did you mean my two-octave semitenth or something else?

Petr

--- In tuning@yahoogroups.com, Petr Pařízek <petrparizek2000@...> wrote:

> [-1 -11 -1 0 6]
>
> This suggests a "sevenless" 3D temperament of unmatched accuracy and rarely
> low complexity in the realm of microtemperaments. I'm only not sure about
> the generators right now.

The planar temperament can use 2, 3/2 and 11/9 as generators. Another possibility would be 2, 11/9 and 243/242.

2011/7/11 Petr Pařízek <petrparizek2000@...>
>
> Mike wrote:
>
> > In the 5-limit, this is interesting; it continues the pattern magic,
> > wurrschmidt, parizek#2. Magic combines the chain of thirds with 3/1,
> > Wurrschmidt does it with 6/1, this seems to do it with 12/1.
>
> The 11th root of 12/1, if that's what you're talking about, is remarcably
> wider than 5/4 and therefore I don't see much of a point in such a
> temperament, concerning its complexity. For that reason, I'm not surprised I
> haven't seen this temperament mentioned anywhere.
>
> How have you found those patterns?
>
> BTW: By "parizek#2", did you mean my two-octave semitenth or something else?

Sorry, I realize now that you were posting monzos. I thought you were
writing generator mappings for some reason.

-Mike